Glossary

8.2 Tangent spaces and Jacobian criterion

4 min read•july 30, 2024

Tangent spaces are crucial for understanding the local geometry of algebraic varieties. They provide a linear approximation of the variety at a point, helping us analyze smoothness and detect singularities. This concept is fundamental in algebraic geometry.

The is a powerful tool for detecting and classifying singularities. By examining the rank of the , we can determine whether a point is smooth or singular, shedding light on the variety's local structure and overall geometry.

Tangent spaces on varieties

Computing tangent spaces

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  • The at a point P on an V is the vector space of all tangent vectors to V at P
  • For an affine variety V defined by polynomials f1,...,fkf_1, ..., f_k in nn variables, the tangent space at a point P is the kernel of the Jacobian matrix J(f1,...,fk)J(f_1, ..., f_k) evaluated at P
    • The Jacobian matrix contains the of the defining polynomials with respect to each variable
    • The kernel of the Jacobian matrix represents the directions in which the variety is "flat" or has zero curvature at P
  • The tangent space can be computed by finding the nullspace of the Jacobian matrix using techniques from linear algebra, such as Gaussian elimination or singular value decomposition

Properties of tangent spaces

  • The dimension of the tangent space at a is equal to the dimension of the variety
    • A smooth point is a point where the variety is locally a manifold, without any singularities or self-intersections
  • For a projective variety, the tangent space is defined in the same way, but using homogeneous coordinates
    • Homogeneous coordinates allow the representation of points at infinity and provide a consistent framework for studying projective varieties
  • The tangent space provides a linear approximation to the variety near the point P, which is useful for studying the local geometry and singularities of the variety

Singularity detection with the Jacobian

Jacobian criterion for singularities

  • A point P on an algebraic variety V is singular if the dimension of the tangent space at P is greater than the dimension of V
    • Singular points are points where the variety is not locally a manifold, such as self-intersections, cusps, or double points
  • The Jacobian criterion states that a point P on an affine variety V defined by polynomials f1,...,fkf_1, ..., f_k is singular if and only if the rank of the Jacobian matrix J(f1,...,fk)J(f_1, ..., f_k) evaluated at P is less than the codimension of V (n−dim⁡(V)n - \dim(V))
    • The codimension of V is the difference between the ambient space dimension nn and the dimension of the variety dim⁡(V)\dim(V)
  • For a hypersurface defined by a single polynomial ff, a point P is singular if and only if all partial derivatives of ff vanish at P
    • A hypersurface is a variety of codimension 1, such as a curve in a plane or a surface in 3-dimensional space

Classifying singularities

  • The Jacobian criterion can be used to classify the types of singularities, such as nodes, cusps, and double points, based on the rank of the Jacobian matrix and the local geometry of the variety near the
    • A node is a singular point where two branches of the variety intersect transversely (example: the origin of the curve y2=x2(x+1)y^2 = x^2(x+1))
    • A cusp is a singular point where two branches of the variety meet tangentially (example: the origin of the curve y2=x3y^2 = x^3)
    • A double point is a singular point where the variety intersects itself with multiplicity 2 (example: the origin of the curve y2=x2y^2 = x^2)
  • The classification of singularities helps understand the local structure of the variety and is important in the study of resolution of singularities and the geometry of algebraic curves and surfaces

Jacobian matrix: Geometry and rank

Geometric interpretation of the Jacobian matrix

  • The Jacobian matrix J(f1,...,fk)J(f_1, ..., f_k) encodes the first-order approximation of the algebraic variety V defined by polynomials f1,...,fkf_1, ..., f_k near a point P
    • The first-order approximation is the linear part of the Taylor expansion of the defining polynomials at the point P
  • The kernel of the Jacobian matrix at a point P corresponds to the directions in which the variety is "flat" or has zero curvature at P
    • These directions form the tangent space to the variety at P
  • The image of the Jacobian matrix at a point P corresponds to the directions in which the variety is "curved" or has non-zero curvature at P
    • These directions are orthogonal to the tangent space and indicate how the variety bends or twists near P

Rank of the Jacobian matrix and local geometry

  • The rank of the Jacobian matrix at a point P determines the dimension of the tangent space and the local geometry of the variety near P
    • The rank is the number of linearly independent rows or columns of the Jacobian matrix
  • If the rank of the Jacobian matrix is equal to the codimension of V, then the point P is a smooth point, and the variety is locally a manifold near P
    • At a smooth point, the tangent space has the same dimension as the variety, and the variety has no singularities or self-intersections
  • If the rank of the Jacobian matrix is less than the codimension of V, then the point P is a singular point, and the variety has a more complicated local structure, such as self-intersections or cusps
    • At a singular point, the tangent space has a higher dimension than the variety, indicating the presence of multiple tangent directions or a degenerate structure
  • The rank of the Jacobian matrix can be computed using techniques from linear algebra, such as Gaussian elimination or singular value decomposition, and provides crucial information about the singularities and the local geometry of the variety

Key Terms to Review (16)

Algebraic variety: An algebraic variety is a fundamental concept in algebraic geometry, representing a geometric object defined as the solution set of polynomial equations. These varieties can be classified into affine varieties, which are subsets of affine space, and projective varieties, which exist within projective space. Understanding algebraic varieties helps in studying their properties, including dimensions, singularities, and their relationships to ideals and polynomial rings.
Dimension of tangent space: The dimension of tangent space refers to the number of independent directions in which one can move away from a point on a manifold or algebraic variety. It provides a way to understand the local behavior of a geometric object at a specific point, reflecting how many parameters are needed to describe nearby points. This concept is crucial when analyzing smoothness and singularities within algebraic geometry and is tied to the Jacobian criterion, which helps determine the dimensions of these spaces in relation to the underlying structure.
Embedding Dimension: The embedding dimension of a variety is the minimum number of dimensions in which it can be realized as a subvariety. This concept helps in understanding how geometric objects can be situated in higher-dimensional spaces, revealing properties about their local structure and behavior. Analyzing the embedding dimension also links to the study of tangent spaces, which are crucial for assessing the local properties of varieties and applying the Jacobian criterion.
Geometric Tangent Space: The geometric tangent space at a point on a variety is a vector space that intuitively represents the directions in which one can move from that point within the variety. It captures the local linear structure of the variety, helping to analyze how functions behave near that point. This concept is crucial in understanding curves and surfaces within algebraic geometry, as it allows mathematicians to study properties such as singularities and intersections more effectively.
Jacobian Criterion: The Jacobian Criterion is a mathematical tool used to determine the smoothness of a variety at a given point. It assesses whether the intersection of the variety with a given set of equations is smooth by examining the rank of the Jacobian matrix, which consists of partial derivatives of the defining functions. This criterion plays a crucial role in understanding tangent spaces and singularities within algebraic geometry.
Jacobian Matrix: The Jacobian matrix is a matrix of first-order partial derivatives of a vector-valued function. It plays a crucial role in understanding how changes in input variables affect output variables, particularly in multivariable calculus and algebraic geometry. The Jacobian is essential for identifying regular and singular points, as well as for analyzing tangent spaces and applying the Jacobian criterion for determining the nature of critical points.
Local behavior: Local behavior refers to the properties and characteristics of mathematical objects, such as functions or varieties, in a small neighborhood around a point. It provides insight into how these objects behave near specific points, which is crucial in understanding their overall structure and properties. In algebraic geometry, analyzing local behavior helps in studying tangent spaces and applying the Jacobian criterion to assess singularities and smoothness.
Local study of varieties: The local study of varieties involves analyzing algebraic varieties in a localized context, often focusing on their behavior near specific points. This approach is essential for understanding the geometric and algebraic properties of varieties by examining tangent spaces and singularities. It provides tools to investigate how varieties behave in small neighborhoods around points of interest, thus revealing deeper insights into their structure and characteristics.
Non-singular variety: A non-singular variety is a geometric object in algebraic geometry that has no 'sharp points' or 'edges', meaning it is smooth and well-behaved everywhere. This quality implies that the variety has a well-defined tangent space at every point, which leads to important implications for its local and global properties, including the ability to apply various mathematical tools effectively.
Partial Derivatives: Partial derivatives measure how a function changes as one of its input variables changes while keeping the other variables constant. This concept is essential in understanding functions of multiple variables, and it plays a critical role in analyzing the behavior of these functions in higher-dimensional spaces, particularly when looking at tangent spaces and applying the Jacobian criterion.
Rank Condition: The rank condition is a criterion used to determine the dimensionality of the tangent space at a point on an algebraic variety. It involves analyzing the rank of the Jacobian matrix formed by the partial derivatives of defining equations, and it plays a crucial role in understanding singularities and smoothness of varieties. When the rank condition is satisfied, it indicates that the variety is smooth at that point, while failure to meet this condition suggests the presence of singular points.
Singular point: A singular point is a point on a geometric object where the object fails to be well-behaved in some way, such as having a cusp or a node. These points often indicate a breakdown in the smoothness or differentiability of the object, making them essential for understanding the overall structure and behavior of curves and surfaces. Recognizing and analyzing singular points is crucial for determining the properties and classifications of various geometric forms.
Smooth point: A smooth point on a variety is a point where the tangent space has the expected dimension, meaning that the local behavior of the variety is well-behaved and resembles that of a manifold. At a smooth point, the Jacobian matrix, which consists of the partial derivatives of defining equations, has full rank, ensuring that there are no singularities or abrupt changes in the structure of the variety around that point.
Smoothness analysis: Smoothness analysis is the study of how a geometric object behaves at a point, particularly whether the object has well-defined tangent spaces at that point. This concept is crucial for understanding the local properties of varieties, which helps determine if they are smooth or singular based on their derivatives and behavior under perturbations.
Tangent space: The tangent space at a point on a variety is a vector space that represents the possible directions in which one can tangentially pass through that point. It provides a linear approximation of the variety at that specific point, reflecting how the variety behaves locally. Understanding tangent spaces is crucial for studying the geometry and smoothness of varieties, as well as for applying tools like the Jacobian criterion to analyze singularities and local properties.
Tangent Vector: A tangent vector is a mathematical object that represents a direction and rate of change at a specific point on a curve or surface. It serves as a foundational concept in calculus and differential geometry, allowing us to analyze how functions behave locally. The idea of tangent vectors is crucial for understanding properties such as smoothness, differentiability, and the geometry of curves and surfaces, particularly in the study of tangent spaces and the Jacobian criterion.
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